For questions about or involving covering spaces.

learn more… | top users | synonyms

7
votes
0answers
2k views

The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent

This is exercise 1.3.8 in Hatcher: Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then ...
6
votes
0answers
129 views

Are there infinitely many non-negative integers not covered by one of these 7 polynomials?

Consider the following polynomials: $$ \begin{align} f_1(n, m) &= 30nm + 23n + 7m + 5 \\ f_2(n, m) &= 30nm + 17n + 13m + 7 \\ f_3(n, m) &= 30nm + 23n + 11m + 8 \\ f_4(n, m) &= ...
5
votes
0answers
127 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
5
votes
0answers
78 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
5
votes
0answers
45 views

How to construct certain cover given in Mumford's Abelian Varities book

In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
4
votes
0answers
36 views

Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over ...
4
votes
0answers
109 views

An entire function with finite covering group is a polynomial.

Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
3
votes
0answers
57 views

Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
3
votes
0answers
40 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
3
votes
0answers
58 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
3
votes
0answers
92 views

Connected Components of Covering Space

Let $\pi : \tilde M\to M$ be a covering map, $K_1, K_2\subset\tilde M$ are two different connected components and there exists such points $x\in K_1, y\in K_2$ that $\pi(x)=\pi(y)$. In other words, ...
3
votes
0answers
84 views

deck transformations and covering spaces

Let $p:\tilde X\rightarrow X$ be a universal covering space, and let $H\leq G$ where $G$ is the group of covering transformations. Let $q:\tilde X \rightarrow \tilde X/G$ be the quotient map which is ...
3
votes
0answers
377 views

Algorithms for covering a rectilinear polygon using rectangles of the same size

The following is the problem description: All angles of the polygon are right. It may be convex or concave. Use rectangles of the same size to cover the polygon. The edge of the polygon and ...
2
votes
0answers
56 views

Branched coverings of unit disk

Is there a classification of branched coverings of the closed unit disk $\mathbb{D} =\{z\in \mathbb{C} \ | \ |z| \leq 1 \}$? Here we consider only branched covering projections which restrict to ...
2
votes
0answers
76 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if ...
2
votes
0answers
33 views

Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
2
votes
0answers
42 views

lifting injective maps to injective maps in principal bundles

Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde ...
2
votes
0answers
23 views

Correct definition of a regular covering without global connectedness hypotheses

Let $p:Y\to X$ be a covering map of topological spaces where $X$ is assumed to be locally path connected (and hence the same is true of $Y$) but neither $X$ nor $Y$ is assumed to be connected. In this ...
2
votes
0answers
22 views

Submersions and complex structure

Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ ...
2
votes
0answers
40 views

Covering Spaces and Fundamental Groups

Can somebody tell me if what I did is right? I need to Draw the based cover $\hat{B}\rightarrow B$ such that $\pi_{1}(\hat{B},v)$ corresponds to the subgroup $\langle a^{3}, a^{2}b\rangle$ ...
2
votes
0answers
101 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
2
votes
0answers
32 views

Degree and picture of a Covering map

I need to know if I am right: I need to know the degree of this covering map $R \rightarrow S$: $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ I have that genus of $R$, $g_{R} ...
2
votes
0answers
38 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
2
votes
0answers
85 views

Fundamental group and path-connected

Let $p:E \rightarrow B$ be a covering space, $E$ and $B$ are path-connected. Let $A$ be a path-connected subset of $B$ . How to use fundamental group to give a sufficient and necessary condition to ...
2
votes
0answers
53 views

Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
2
votes
0answers
167 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then ...
2
votes
0answers
63 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
2
votes
0answers
108 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
2
votes
0answers
97 views

Characterization of maps with $\mathbb{Z}/2\mathbb{Z}$-equivariant lifts.

I'm interested in characterizing maps $f:\mathbb{R}P^k\to\mathbb{R}P^\infty$ that lift to a $\mathbb{Z}/2\mathbb{Z}$-equivariant map $\tilde{f}:S^k\to S^\infty$. For $k\geq 2$ I have been able to ...
2
votes
0answers
32 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
2
votes
0answers
59 views

Global sections of covering spaces

Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$. Is there any reference where this is explicitly ...
2
votes
0answers
93 views

Are there infinitely many rational functions of bounded degree and given ramification

It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus is finite (up to isomorphism). Edit. The branch locus $B$ of $f:X\to ...
2
votes
0answers
103 views

Identifying the numbers of degree $n$ covering spaces of $X$

Let $X$ be a path-connected, locally path-connected and semilocally simply-connected space. Can we find a correspondence between degree $n$ covering spaces of $X$ and group homomorphism ...
2
votes
0answers
240 views

Monodromy Theorem and Homotopy Lifting Theorem

I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct? Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a ...
2
votes
0answers
71 views

Liftings of curves $u\cdot v$ and $v\cdot u$ with respect to the sine covering map.

I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache: Let $X=\mathbb{C}\setminus\{\pm1\}$ and $Y ...
2
votes
0answers
53 views

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...
2
votes
0answers
130 views

Hyperbolic Universal Covering Space

I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
1
vote
0answers
22 views

Covering a finite collection intervals

I was trying to solve one of the problems in stein's real analysis book Suppose $I_1, I_2, . . . , I_N$ is a given finite collection of open intervals in R. Then there are two finite sub-collections ...
1
vote
0answers
52 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
1
vote
0answers
47 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
1
vote
0answers
52 views

Show That a Lift Always Exists

I've been considering this problem: Suppose that $X$ is a topological space and that $H_1(X)$ is a finite group of odd order. Show that if $p:\tilde{Y}\rightarrow Y$ is a covering space of index ...
1
vote
0answers
92 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
1
vote
0answers
31 views

Are two equivalent coverings of a Riemann surface also biholomorphic?

Consider a compact Riemann surface $X$. If $p_1:Y\longrightarrow X $ and $p_2:Z\longrightarrow X$ are two topological coverings of $X$, then it is univocally defined a structure of Riemann surface on ...
1
vote
0answers
48 views

Is a covering space of a completely regular space also completely regular

I'm trying to solve a problem in Munkres' Topology book. Let $p: E \rightarrow B$ be a covering map and suppose that $B$ is completely regular (for any closed subset $A$ and disjoint point $a$ there ...
1
vote
0answers
37 views

Cover polygon with rectangles

I need to cover some polygon with rectangles here's an example : The black figure in a black square is the polygon that i need to cover with those green rectangles but i need to do it more ...
1
vote
0answers
22 views

Covering spaces and Automorphisms

I need to find for the groups $G$ a connected degree-4 cover $\hat{B}\rightarrow B$ such that Aut($\hat{B}\rightarrow B$) is isomorphic to $G$ $G \cong 1$ $G \cong \mathbb{Z}_{2}$ $G \cong ...
1
vote
0answers
68 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
1
vote
0answers
41 views

Group of covering transformations

The group of automorphisms of a covering $p: E \mapsto X$, to be denoted $Aut(E,p)$, is usually referred to as the group of covering transformations. If $p: E_1 \mapsto E_2$ is an isomorphism of ...
1
vote
0answers
100 views

find the connected covering space of $\mathbb{R}P^2 \lor \mathbb{R}P^2$

This is a problem on Hatcher' book. How to find all the connected covering spaces of $\mathbb{R}P^2 \lor \mathbb{R}P^2$? I don't know where to start. Is there a general way to construct the ...
1
vote
0answers
66 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...